Pauli first noticed the hidden SO(4) symmetry for the Hydrogen atom in the early stages of quantum mechanics [1]. Departing from that symmetry, one can recover the spectrum of a spinless hydrogen atom and the degeneracy of its states without explicitly solving Schr\"odinger's equation [2]. In this paper, we derive that SO(4) symmetry and spectrum using a computer algebra system (CAS). While this problem is well known [3, 4], its solution involves several steps of manipulating expressions with tensorial quantum operators, simplifying them by taking into account a combination of commutator rules and Einstein's sum rule for repeated indices. Therefore, it is an excellent model to test the current status of CAS concerning this kind of quantum-and-tensor-algebra computations. Generally speaking, when capable, CAS can significantly help with manipulations that, like non-commutative tensor calculus subject to algebra rules, are tedious, time-consuming and error-prone. The presentation also shows a pattern of computer algebra operations that can be useful for systematically tackling more complicated symbolic problems of this kind.

中文翻译：

---

量子龙格-楞次向量和氢原子，使用计算机代数隐藏的 SO(4) 对称性

在量子力学的早期阶段，泡利首先注意到氢原子隐藏的 SO(4) 对称性 [1]。脱离这种对称性，我们可以恢复无自旋氢原子的光谱及其状态的简并性，而无需明确求解 Schr\"odinger 方程 [2]。在本文中，我们使用计算机推导出 SO(4) 对称性和光谱代数系统 (CAS). 虽然这个问题是众所周知的 [3, 4]，但它的解决方案涉及使用张量量子算子处理表达式的几个步骤，通过考虑组合交换器规则和爱因斯坦的重复索引求和规则来简化它们。因此，它是一个很好的模型来测试 CAS 在这种量子和张量代数计算方面的现状。CAS 可以极大地帮助处理像受代数规则约束的非交换张量演算那样乏味、耗时且容易出错的操作。该演示文稿还展示了一种计算机代数运算模式，可用于系统地解决此类更复杂的符号问题。